Aspects of endomorphism monoids of certain algebras

This thesis is concerned with the study of endomorphism monoids of certain algebras. We first describe the semigroup structure of a family of subsemigroups of the endomorphism monoid of an independence algebra A. Each of these subsemigroups is associated with a subalgebra B of A and is called the subsemigroup of endomorphisms with restricted range in B. Denoted by T(A,B), it consists of all endomorphisms of A whose image lies in B. We show in particular that such semigroups are not regular in general and that they present significant differences in their structure from that of End(A).

In a similar fashion, we investigate the semigroup structure of End(T_n), the endomorphism monoid of the full transformation monoid of a finite set with n elements.
We describe the ideals of End(T_n) and show that, in particular, T_n and End(T_n) are not respectively embeddable into each other (except in the degenerate case of n=1).

We then move on to the study of translational hulls of ideals of the endomorphism monoid of algebras. We start with the case of an independence algebra A, where we discuss the translational hull Ω(I) of the (0-)minimal ideal I of End(A).
We give conditions under which Ω(I) and End(A) are isomorphic and we construct a canonical isomorphism where possible. A more general approach of translational hulls in the case where A is an arbitrary algebra is then presented, where we prove that any ideal I of End(A) satisfying some representability and separability conditions on A will be such that its translational hull is isomorphic to End(A).
Finally, we close this thesis by computing the translational hulls of some of the ideals of End(A), where A will stand either for a free algebra, an independence algebra, or T_n.